Half-angle set

The half-angle rates are formulas of trigonometry , which for special application cases logarithmically useful for determining the determinants (pages a, b, c; angle , , ) of general triangles have been developed. Corresponding theorems apply to general triangles on a spherical surface ( spherical geometry ). ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$

Half-angle blocks in the plane

• ${\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}$
• ${\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}$
• ${\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{s(s-a)}}}}$

in which ${\displaystyle s={\frac {a+b+c}{2}}}$

The equivalent statement to the third formula

• ${\displaystyle \cot {\frac {\alpha }{2}}={\frac {s-a}{\rho }}={\sqrt {\frac {s(s-a)}{(s-b)(s-c)}}}}$

is also known as the cotangent theorem. denotes the incircle radius here . ${\displaystyle \rho }$

Corresponding formulas apply to the other angles.

Half-angle blocks on the spherical surface

• ${\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin b\,\sin c}}}}$
• ${\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {\sin s\,\sin(s-a)}{\sin b\,\sin c}}}}$
• ${\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin s\,\sin(s-a)}}}}$

in which ${\displaystyle s={\frac {a+b+c}{2}}}$